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# 2 Open questions

[2327.2.1] Numerous open questions surround the traditional model above. [2327.2.2] Important examples are the macroscopic effects of capillarity and surface tensions, the spatiotemporal variability of residual saturations, hysteresis and saturation overshoot (see e.g. [1, 36, 37, 18, 38, 39, 40, 24, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50] and references therein). [2327.2.3] In this paper we focus on the interplay and relation between hysteresis and saturation overshoot.

[2327.3.1] Saturation overshoot refers to the experimental observation of non-monotone saturation profiles during certain classes of infiltration problems, where fingers of infiltrating water develop due to gravitational instabilities [51, 52, 53, 54, 55]. [2327.3.2] Within the fingers, the profile of water saturation is non-monotone [53]. [2327.3.3] Experiments in relatively thin tubes, with diameter less than the characteristic finger width, show the existence of similar non-monotone profiles [53, 54, 55].

[2327.4.1] Many theoretical and numerical investigations have in recent years addressed saturation overshoot and gravity driven fingering and their relation to each other [56, 57, 58, 59, 11, 60, 61, 62, 63, 64, 65, 66]. [2327.4.2] Ref. [56] reported saturation overshoot in numerical solutions of Richards equation when the hydraulic conductivities between collocation points weighted appropriately in the numerical scheme. [2327.4.3] In [11] it was argued that these overshoot solutions are numerical artifacts generated by truncation errors. [2327.4.4] This finding agreed with earlier mathematical proofs of existence, uniqueness and stability of solutions within an L1-theory for a class of quasilinear elliptic-parabolic equations [67, 32, 57, 58]. [2327.4.5] These were applied to the Richards equation [32, 61, 68] leading to the conclusion that the conventional theory is inadequate to represent fingering and overshoot [61, 68, 63, 69, 70].

[2327.5.1] Accordingly, new approaches were proposed [60, 63, 64, 65, 48, 71, 72]. Ref. [60] discussed additional terms in Richards’ equation, meant to represent a so called ’hold-back-pile-up’ effect. [2327.5.2] Other suggestions are based on dynamic extensions of capillary pressure [71, 63, 72]. [2327.5.3] Some authors proposed an additional term based on the yet to be observed “effective macroscopic surface tension” [64, 65]. [2327.5.4] Saturation overshoot profiles at rest when the velocities of all fluids vanish were recently predicted within an approach based on distinguishing percolating and nonpercolating (trapped, immobile) fluid parts in [48].

[2327.6.1] The overshoot phenomenon continues to challenge researchers in the field. [2327.6.2] There is an ongoing and lively scientific debate on the subject as witnessed by numerous publications (see also [70] for more references). [2327.6.3] Very recently, DiCarlo and coworkers [69] investigated the physics behind the displacement front using the traditional model of two phase flow. [2327.6.4] They argue that gas is drawn in behind the overshoot tip and that the viscosity of this gas plays an observable role when the infiltrating flux is large. [2327.6.5] They emphasize, however, that in their paper they are “not concerned with why the overshoot occurs, or in other words why the saturation jumps” and that “adding in the viscosity of the [page 2328, §0]    displaced phase…does not change the arguments” of van Duijn et al. [63]. [2328.0.1] According to these arguments the traditional standard model leads to parabolic equations and hence overshoot behaviour is not allowed [69, p.965].

[2328.1.1] The objective for the rest of this paper is to review analytical and numerical evidence for the possible existence of non-monotone solutions (saturation overshoot profiles) within the traditional standard theory outlined above.